As a stylus, a motion-capture target, or another pen-tip moves through a two-dimensional (or a three-dimensional) space, the pen-tip's location can be sampled, thereby providing data points through which a motion path, a digital ink path, and/or another curve can be rendered. Depending on factors such as the speed of the pen-tip, the path followed by the pen-tip, and the sampling rate, the data points may be either more or less accurate in their approximation of the actual path followed. Sometimes that accuracy is the most important criterion for evaluating a curve which is rendered (visually or virtually) through or near the data points. The foregoing observations are made with the present claims in mind. Accordingly, one of skill in the art who did not have access to the claims would not necessarily have recognized the various uses to which data points and rendered curves can be put.